For a similar use of d\Aa pay... ye confirming a view of the opponent’s position which has already been stated ( they not only say So but they must say so’) cf. T. 1007» 20-29 éorat yap 76 aiTo fal Tpunpyns kal Totxos Kal dvOpwros, ei Kara mayTos TH Katapio ou 7 dropinr at évoexerar . . » GXAD py AexTéeov Y adrois Kata mavTds (ravTos) THY Katapacw % THv arodacw.
13. AaBotoa, ‘having received’, not ‘having taken’. The material principle receives the formative principle as the female receives the seed, which is the formal principle of generation. For the analogy cf. A. 987> 339882 ¥, and for the literal sense of AapPavew cf. H. A. 559° 8, 577° 31, 32, 578% 14, 6329 28.
17-20, The union might be of the nature (a) of the accidental union of a subject with an attribute, or (0) of the intimate and essential union of genus with differentia, For the difference cf. Z. 1030% 11-14, 1037 13-21.
17. The manuscript reading pebééer Oarépov Garepov (‘ one will partici- pate in the other’) does not offer a grammatical parallel to the other alternative 7 dray 7 xrA. Christ is therefore right in proposing peOeger Barépou Sarépou (‘ by participation of one in the other’).
18. petéxer yap todtwy. We should expect peréxer yap 6 avOpwros Tod AevKod, answering to pebeée Garépov Oarépov. But Aristotle is not careful about consistency of expression where his general meaning is clear.
20-21. As instances of things which are ev addy, pige, Pere. Alexan- der mentions a bundle of sticks, mead, and the stones in a house. igus means complete fusion ; the distinction between ady and Ogois is not so clear, but probably ag@y means mere contact which may be accidental, while @éo1s does not necessarily imply contact but does imply intentional arrangement. Cf. H. 1042> 15-20.
30-31. ‘As the definite or ideal 2 produced (by co-operation with the indefinite dyad) the 2’s in 4, so these 2’s (again by co-operation with the indefinite dyad) produced the two 4’s (more strictly, the four 2's) in 8.
gi—>1. The argument in ll. 26-31 was directed to showing that some 2’s are prior to others. But in the course of the argument Aristotle showed that the 2’s in 4 discharge a function analogous to that of the ideal 2. He now draws another inference from this, viz. that there- fore they also must be Ideas. And so, too, will be the units in the ideal 2 which similarly (by co- operation with the indefinite dyad) pro- duce the units in 4. Thus the units in any number are Ideas.. Ideas will be composed of Ideas. The Idea of one animal will contain the
Ideas of other animals, and therefore one animal itself will contain other animals.
Jaeger points out that E1JA» read idea: after duds in 1. 32 (he infers from Al, 758. 21 that Alexander also read id€a; but 758. 25 has idéa). He concludes that 4 zpury terpas has dropped out before kai 7 mporn duds. But the argument in Il. 28-32 is that the dyads in 4, since they generate the tetrads in 8 as the first dyad generates them, must be Ideas as much as the first dyad; a reference to the first tetrad would be out of place. id¢a: has come in because the eye of the writer of the archetype travelled on to idéa: later in the line. For the idiomatic xaé before 7 mparn duds cf. N. 1089® 16.
bi, et toUtwy iSéa. eiciv. Christ’s suspicion of these words seems quite unfounded, and it seems clear that Alexander read them (759. 4). They may be taken in either of two ways. -(1) We may render ‘if there are Ideas of animals’, or (2) we may take ofov . . . Ggwy as parenthetical and render ‘if the Ideas are Ideas of the sensible things’.
6. povadixor, cf. ro80b1gn, Aristotle means that while as regards two concrete groups we might find some difficulty in admitting the disjunction ‘they are either equal or unequal’, we can find no difficulty about two abstract numbers.
11-16, From the premise that if we add one unit to another we always get a 2, two objections to the view here attacked follow : (1) the 2 thus formed will be composed of units specifically different, which contradicts the view in question; and (2) it will be hard for the thinkers in question to say whether it is prior or posterior to the 3. * It would seem more likely to be prior, since one of its elements is prior to and the other simultaneous with the 3. Why then should the Platonists not say that it is prior? Aristotle does not say why, but no doubt he means that it will be awkward for them thus to put a number. between 2 and 3.
21-22. Sijdov.. . eveore TH Sudds. ‘Clearly there is in 3 a number equal to 2.’
22-23. ‘But the 2 in 3 cannot be equal to the 2 itself, if there is a first and a second number ’, i.e. if 2 and 3 are numbers qualitatively different from one another. Cf. 1080417, 19, where 70 ev mparov te abrod (i. €. rod dpiOuod) 7d & exdpevov is synonymous with dovpBAyros.
23-24. ‘Nor will the Ideas be numbers,’ sc. as common sense understands numbers.
26. mpdtepov, 10813 5-17.
28-30. ‘ For which reason they must say that when we count thus, “1, 2”, we do not do this by adding 1 to the previous 1.’
32. mdvra Ta eldy Evds pepy, i.e. all the Forms would be parts of the Form which is the largest number.
34-37. Alexander’s commentary (762. 17—763. 3) may be sum- marized thus: ‘ The Platonists raise this difficulty, whether when we count “1, 2”’ we count by addition or by successive divisions of ro. They say we cannot do so in either way; not in the former because
440 Commentary
then we should be treating the units as comparable, and not in the latter because then we should make the Idea of 10 contain the Ideas of the smaller numbers. We must confine both addition and division to mathematical number, and recognize ideal. number as otherwise produced. But we actually count, says Aristotle, in both ways. If the number is definite, like 8, we divide it into its proper parts; if it is indefinite we add unit to unit till we reach the number we wish to determine. But what does he mean by indefinite number? Perhaps he’means that the numbers included in 20 are indefinite and of the nature-of matter relatively to 20. And so, he says, it is absurd on the strength of this superficial dzopia to say that each of the numbers is a separate Idea and substance’. Bz. seems right in supposing that Alexander -had before him words which do not exist in our text (cf. especially 762. 32 d\AG w&s aopicrov etre Tov dpiOuov;); and there are traces of something similar in Syr.
As regards xara pepidas Alexander’s interpretation is probably guesswork. A better interpretation has been suggested by Apelt. He quotes, for the meaning of pepis, Plut. Quaest. Conv, il. 2.644 C Ta Sdypooa detrva pos pepida yiyverGa, which means ‘to be served separately by portions, so that each guest has his separate dish’. Aristotle’s point seems to be this: The Platonists deny many of the accepted truths of mathematics (for roAAa dvaipotow cf. 1086% 9). E.g. they think they will put us in a difficulty by asking us whether we count by addition or by separate portions, i.e. constructing each number independently. If we say ‘by addition’, they will answer ‘then you are not grasping the nature of abstract number’; if we say ‘by separate portions’, they will answer ‘then you are already admitting a number other than mathematical number’. But in fact we can regard the process in either way; it is absurd to rest the doctrine of two entirely different kinds of number on so superficial an dzropia.
10837 1-2. Aristotle’s own view is that numbers have a differentia, but units have not.
4. The grammar requires émdpxew (which Alexander seems to have probably read, 763. 9) instead of the manuscript reading brapxov.
GAN’ 7 dpiOuds, KaT& td woody, ‘but number gua number differs in respect of guantily’.
11. By the quality of number Aristotle means such attributes as compositeness ()( primeness) and being ‘plane’ or ‘solid’ (having two or three factors); cf. A. 10203. These attributes, according to Aristotle, attach to a number in virtue of its quantity.
12. THs Suddos, the indefinite dyad.
13. Bz.’s reading wogorovdy is evidently right, though it is read only by Syr. and the second hand of E. The word seems to be a hapax
M. 8. 10832 1 — 1083 20 44h
legomenon, but dvorrows ( 36, 1082%15) supplies an analogy, and the play on words supplies a motive, for the coinage.
17-19. 61 ... pavepdv summarizes 1081 5-17; IQ-20. oUTe.. tTpomwy Summarizes 10819 17—» 35, » 35108317.
20-1, Aristotle here passes from Plato’s views (cf. ].-32) to discuss those of Speusippus. Cf. 10764 20-21 n.
24-27. Aristotle shows that Speusippus was inconsistent in that while retaining as the formal cause of number the One, conceived as a. separate substance and distinguished from mathematical units (cf. Z. 1028» 21 n.), he did not similarly believe in a Two or Three dis- tinguished from the many twos and threes of arithmetic.
32. domep MAdrwv eXeyev is important as showing that the whole discussion in 1080 3y—1083* 17 was a discussion of Plato’s views rather than of those of his followers. _Speusippus is discussed much more briefly in 1083% 20—» 1, and Xenocrates in » 1-8. The imperfect tense indicates that Aristotle is thinking of Plato’s lectures rather than of published works. ‘This is the only reference to Plato by name in MN.
35: cipy tat, 1080P 341083" 17.
2. 6 tptros tpémos, that of Xenocrates, 1080) 22, 23. Aristotle omits here the view of aAAos tis (1080? 21) that ideal number is the only number that exists.
6. pnkivew, cf. N. Togo? 29 core & od xaXerov brovacoty imobécets AapBavovras prakporovety Kal ouvelpeuy. As Robin remarks, Xenocrates’ prolixity on mathematical matters is indicated by the list of his works in Diog. Laert. iv, 2. 13.
8. Aristotle now recurs to the Pythagorean view (1080? 16-21), which agrees with that of Speusippus in believing only in mathematical number, and differs from all the Platonic views in regarding numbers as the stuff out of which things are made.
8-9. Ti pev... cipnpévav. The Pythagorean view avoids the mis- takes implied in separating the substance of a thing from the thing itself (Z. 6).
13-19. The argument is as follows:
There are no indivisible magnitudes (proved in De Gen. et Corr. 315” 24—3177 17).
Or at least units (numerical indivisibles) have not magnitude.
And a magnitude cannot consist of indivisibles.
But arithmetical number consists of units, which are indivisibles.
Therefore real things, which are magnitudes, cannot consist of numbers. ,
But these thinkers apply arithmetical theorems directly to real things, and imply that real things consist of numbers.
Alexander illustrates this by saying that they thought body: in general was composed of ae number 210, fire of the number 11, air of 13, water of 9
20. tav cipypévay tpdmwv, sc. those enumerated in 10804 15—) 36 and refuted in 1080 371083? 19.
442 Commentary
ARGUMENTS AGAINST ALL THEORIES OF SELF-SUBSISTENT NUMBER (ch. 8. 1083 23—9. 1085? 34).
(1) How are the numbers produced from-the matertal principle ?
1083 23. Is (a) each unit derived from the great and the small, equalized, or (4) one from the small, another from the great? If (6), then (a) the elements are not all present in everything ; (8) the units are not without difference in nature; (y) what of the odd unit in 3? Perhaps this is why they make the One itself occupy the middle place in odd numbers.
30. If (2), then (2) How will 2 be a single entity? How will it differ from a unit? (8) The unit is prior to the two and therefore must be an Idea of an Idea. And it must have been generated before the two. From what, then? Not from the indefinite dyad, for this makes not units but twos.
(2) How many tdeal numbers are there P
36. Number must be either infinite or finite, if it is self-subsistent. But (a) it cannot be infinite. For (a) infinite number is neither odd nor even, but the generation of numbers whether by addition or by multiplication is always either of odd or of even numbers.
1084* 7. (8) If every Idea is an Idea of something, and the numbers are Ideas, infinite number will be an Idea of something, which is neither possible on their theory nor reasonable in itself.‘